General Topology
New submissions
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New submissions for Tue, 28 May 24
- [1] arXiv:2405.16977 [pdf, ps, other]
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Title: All iterated function systems are Lipschitz up to an equivalent metricAuthors: Michał PopławskiSubjects: General Topology (math.GN); Dynamical Systems (math.DS)
A finite family $\mathcal{F}=\{f_1,\ldots,f_n\}$ of continuous selfmaps of a given metric space $X$ is called an iterated function system (shortly IFS). In a case of contractive selfmaps of a complete metric space is well-known that IFS has an unique attractor \cite{Hu}. However, in \cite{LS} authors studied highly non-contractive IFSs, i.e. such families $\mathcal{F}=\{f_1,\ldots,f_n\}$ of continuous selfmaps that for any remetrization of $X$ each function $f_i$ has Lipschitz constant $>1, i=1,\ldots,n.$ They asked when one can remetrize $X$ that $\mathcal{F}$ is Lipschitz IFS, i.e. all $f_i's$ are Lipschitz (not necessarily contractive), $ i=1,\ldots,n$. We give a general positive answer for this problem by constructing respective new metric (equivalent to the original one) on $X$, determined by a given family $\mathcal{F}=\{f_1,\ldots,f_n\}$ of continuous selfmaps of $X$. However, our construction is valid even for some specific infinite families of continuous functions.
Cross-lists for Tue, 28 May 24
- [2] arXiv:2405.16565 (cross-list from math.AC) [pdf, ps, other]
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Title: Invertibility in nonassociative ordered rings and in weak-quasi-topological nonassociative ringsComments: 18 pagesSubjects: Commutative Algebra (math.AC); General Topology (math.GN)
Invertibility is important in ring theory because it enables division and facilitates solving equations. Moreover, rings can be endowed with extra ''structure'' such as order and topology that add new properties. The two main theorems of this article are contributions to invertibility in the context of ordered and weak-quasi-topological rings. Specifically, the first theorem asserts that the interval $]0,1]$ in any suitable partially ordered ring consists entirely of invertible elements. The second theorem asserts that if $f$ is a norm from a ring to a partially ordered ring endowed with interval topology, then under certain conditions, the subset of elements such that $f(1-a) < 1$ consists entirely of invertible elements. The second theorem relies on the assumption of sequential Cauchy completeness of the topology induced by the norm $f$, which as we recall, takes values in an ordered ring endowed with the interval topology (an example of a coarse topology). The fact that a ring endowed with the topology associated with a seminorm into an ordered ring endowed with the interval topology is a locally convex quasi-topological group with an additional continuity property of the product is dealt with in a separate section. A brief application to frame theory is also included.
- [3] arXiv:2405.16592 (cross-list from math.CO) [pdf, other]
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Title: Knot theory and cluster algebras II: The knot clusterComments: 49 pages, 24 figuresSubjects: Combinatorics (math.CO); General Topology (math.GN); Representation Theory (math.RT)
To every knot (or link) diagram K, we associate a cluster algebra A that contains a cluster x with the property that every cluster variable in x specializes to the Alexander polynomial of K. We call x the knot cluster of A. Furthermore, there exists a cluster automorphism of A of order two that maps the initial cluster to the cluster x.
We realize this connection between knot theory and cluster algebras in two ways. In our previous work, we constructed indecomposable representations T(i) of the initial quiver Q of the cluster algebra A. Modulo the removal of 2-cycles, the quiver Q is the incidence quiver of the segments in K, and the representation T(i) of Q is built by taking successive boundaries of K cut open at the i-th segment. The relation to the Alexander polynomial stems from an isomorphism between the submodule lattice of T(i) and the lattice of Kauffman states of K relative to segment i.
In the current article, we identify the knot cluster x in A via a sequence of mutations that we construct from a sequence of bigon reductions and generalized Reidemeister III moves on the diagram K. On the level of diagrams, this sequence first reduces K to the Hopf link, then reflects the Hopf link to its mirror image, and finally rebuilds (the mirror image of) K by reversing the reduction. We show that every diagram of a prime link admits such a sequence.
We further prove that the cluster variables in x have the same F-polynomials as the representations T(i). This establishes the important fact that our representations T(i) do indeed correspond to cluster variables in A. But it even establishes the much stronger result that these cluster variables are all compatible, in the sense that they form a cluster.
We also prove that the representations T(i) have the following symmetry property. For all vertices i,j of Q, the dimension of T(i) at j is equal to the dimension of T(j) at i. - [4] arXiv:2405.16992 (cross-list from math.GR) [pdf, ps, other]
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Title: On non-topologizable semigroupsAuthors: Oleg GutikComments: 9 pagesSubjects: Group Theory (math.GR); General Topology (math.GN)
We find anti-isomorphic submonoids $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$ of the bicyclic monoid $\mathscr{C}(a,b)$ with the following properties: every Hausdorff left-continuous (right-continuous) topology on $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) is discrete and there exists a compact Hausdorff topological monoid $S$ which contains $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) as a submonoid. Also, we construct a non-discrete right-continuous (left-continuous) topology $\tau_p^+$ ($\tau_p^-$) on the semigroup $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) which is not left-continuous (right-continuous).
Replacements for Tue, 28 May 24
- [5] arXiv:2405.01860 (replaced) [pdf, ps, other]
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Title: Characterizing Lipschitz images of injective metric spacesSubjects: General Topology (math.GN)
- [6] arXiv:2405.09437 (replaced) [pdf, ps, other]
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Title: $Γ(X)$ as Metric SpaceAuthors: Edwar Alexis Ramírez ArdilaComments: 13 pages, 3 figuresSubjects: General Topology (math.GN)
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